Journal of applied mathematics & informatics

한국전산응용수학회 (The Korean Society for Computational and Applied Mathematics)
권호 표지

EFFECT OF MATURATION AND GESTATION DELAYS IN A STAGE STRUCTURE PREDATOR PREY MODEL

  • 투고 : 2010.02.09
  • 심사 : 2010.03.27
  • 발행 : 2010.09.30
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초록

In this paper, a stage-structured predator prey model (stage structure on prey) with two discrete time delays has been discussed. The two discrete time delays occur due to maturation delay and gestation delay. Linear stability analysis for both non-delay as well as with delays reveals that certain thresholds have to be maintained for coexistence. Numerical simulation shows that the system exhibits Hopf bifurcation, resulting in a stable limit cycle.

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참고문헌

  1. F. Scudo and J. Ziegler, The Golden Age of Theoretical Ecology : 1923-1940, Springer-Verlag, New York, 1978.
  2. A.G. MacKendrick, Applications of mathematics to medical problems, Proceedings of Edinberg Mathematical Society 44(1926), 98-130.
  3. L. Nielsen, Effect of walleye predation on juvenile mortality and recruitment of yellow perch in oneida lake. Canadian Journal of Fish and Aquatic Science 37(1980), 11-19. https://doi.org/10.1139/f80-002
  4. D. Mech, The wolves of Isle Royale, U.S. Govt. Printing Office, 1966.
  5. G.E. Hutchinson, Circular casual systems in ecology, Ann. N.Y. Acad. Sci. 50(1948), 221-246.
  6. R.M. May, Time-delay versus stability in population models with two and three trophic levels, Ecology 54, 315-325.
  7. S. Blythe, R.M. Nisbet and W. Gurney, Instability and complex dynamica behaviour in population models with long time delay, Theoretical Polulation Biology 22(1982), 147-176. https://doi.org/10.1016/0040-5809(82)90040-5
  8. J. Cushing and M. Saleem, A predator prey model with age structure, Journal of Mathematical Biology 14(1982), 231-250. https://doi.org/10.1007/BF01832847
  9. A. Hasting and D. Wollkind, Age structure in predator-prey systems. i. a general model and a specific example, Theoretical Polulation Biology 21(1982), 44-56. https://doi.org/10.1016/0040-5809(82)90005-3
  10. A. Hasting and D. Wollkind, Age structure in predator-prey systems. ii. functional response and stability and the paradox of enrichment, Theoretical Polulation Biology 21(1982), 57-68. https://doi.org/10.1016/0040-5809(82)90006-5
  11. A. Hasting, Age dependent predation is not a simple process. i. contin- uous time models, Theoretical Polulation Biology 23(1983), 347-362. https://doi.org/10.1016/0040-5809(83)90023-0
  12. A. Hasting, Delays in recruitment at different trophic levels : Effects on stability, Journal of Mathematical Biology 21(1984), 35-44. https://doi.org/10.1007/BF00275221
  13. L. Nunney, The effect of long time delays in predator-prey systems, Theoretical Polulation Biology 27(1985), 202-221. https://doi.org/10.1016/0040-5809(85)90010-3
  14. L. Nunney, Short time delays in population models : a role in enhancing stability, Ecology 66(1985), 1849-1858. https://doi.org/10.2307/2937380
  15. W. Aiello and H. Freedman, A time delay model of single-species growth with stage structure, Mathematical Biosciences 101(1990), 139-153. https://doi.org/10.1016/0025-5564(90)90019-U
  16. W. Aiello, H. Freedman and J. Wu, Analysis of a model representing stage-structured population growth with stage-dependent time delay, SIAM Journal of Applied Mathematics 52(1992), 855-869. https://doi.org/10.1137/0152048
  17. W. Wang and L. Chen, A predator-prey system with stage structure for predator, Comput. Math. Appl. 33 (8)(1997), 83-91. https://doi.org/10.1016/S0898-1221(97)00056-4
  18. X. Zhang, L. Chen and U.A. Neumann, The stage-structured predator-prey model and optimal harvesting policy, Mathematical Biosciences 168(2000), 201-210. https://doi.org/10.1016/S0025-5564(00)00033-X
  19. X. Song and L. Chen, Optimal harvesting and stability for a two-species competitive system with stage structure, Mathematical Biosciences 170(2001), 173-186. https://doi.org/10.1016/S0025-5564(00)00068-7
  20. X. Song and L. Chen, Optimal harvesting and stability for a predator-prey system with stage structure, Acta Math. Appl. Sin. 18 (3)(2002), 423-430. https://doi.org/10.1007/s102550200042
  21. M. Bandyopadhyay and Sandip Banerjee, A stage-structured preypredator model with discrete time delay, Applied Mathematics and Computation 182(2006), 1385-1398. https://doi.org/10.1016/j.amc.2006.05.025
  22. O. Arino, E. Sanchez and A. Fathallah, State-dependent delay differential equations in population dynamics: modeling and analysis, Amer. Math. Soc., Providence, RI 29(2001), 19-36.
  23. R. Xu, M.A. Chaplin and F.A. Davidson, Persistence and stability of a stage-structured predator-prey model with time delays, Appl. Math. Comp. 150(2004), 259-277. https://doi.org/10.1016/S0096-3003(03)00226-1
  24. S.A. Gourley and Y. Kuang, A stage structured predator-prey model and its dependence on maturation delay and death rate, Journal of Mathematical Biology 49(2004), 188-200.
  25. M. Adimy, F. Crauste and S. Ruan, Periodic oscillations in leukopoiesis models with two delays, Journal of Theoretical Biology 242(2006), 288-299. https://doi.org/10.1016/j.jtbi.2006.02.020
  26. Y. Li and Y. Kuang, Periodic solutions in periodic delayed gause-type predator-prey systems, Proceedings of Dynamical Systems and Applications 3(2001), 375-382.